Crinkly Curves

Cantor’s Conundrums

Applications of space-filling curves are necessarily built on finite examples—paths one can draw with a pencil or a computer. But in pure mathematics the focus is on the infinite case, where a line gets so incredibly crinkly that it suddenly becomes a plane.

Cantor’s work on infinite sets was controversial and divisive in his own time. Leopold Kronecker, who had been one of Cantor’s professors in Berlin, later called him “a corrupter of youth” and tried to block publication of the paper on dimension. But Cantor had ardent defenders, too. Hilbert wrote in 1926: “No one shall expel us from the paradise that Cantor has created.” Indeed, no one has been evicted. (A few have left of their own volition.)

Cantor’s discoveries eventually led to clearer thinking about the nature of continuity and smoothness, concepts at the root of calculus and analysis. The related development of space-filling curves called for a deeper look at the idea of dimension. From the time of Descartes, it was assumed that in d-dimensional space it takes d coordinates to state the location of a point. The Peano and Hilbert curves overturned this principle: A single number can define position on a line, on a plane, in a solid, or even in those 11-dimensional spaces so fashionable in high-energy physics.

At about the same time that Cantor, Peano and Hilbert were creating their crinkly curves, the English schoolmaster Edwin Abbott was writing his fable Flatland, about two-dimensional creatures that dream of popping out of the plane to see the world in 3D. The Flatlanders might be encouraged to learn that mere one-dimensional worms can break through to higher spaces just by wiggling wildly enough.